Universality for certain hermitian Wigner matrices under weak moment conditions

www.imstat.org/aihp 2012, Vol. 48, No. 1, 47–79

DOI:10.1214/11-AIHP429

© Association des Publications de l’Institut Henri Poincaré, 2012

Universality for certain Hermitian Wigner matrices under weak moment conditions ¹

Kurt Johansson

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. E-mail:[email protected] Received 26 January 2010; revised 7 April 2011; accepted 11 April 2011

Abstract. We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These ran- dom matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy–Widom universality holds at the edge in this class of random matrices under the optimal mo- ment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

Résumé. Nous étudions l’universalité des statistiques locales du spectre des matrices de Wigner hermitiennes divisibles par une gaussienne. Ces matrices aléatoires sont obtenues en ajoutant à une matrice de Wigner hermitienne avec des coefficients indépen- dants une matrice du GUE indépendante. Nous montrons que la classe d’universalité de la loi de Tracy–Widom pour les valeurs propres extrêmes est vérifiée sous la condition optimale d’une borne uniforme sur le quatrième moment des coefficients de la matrice. De plus, nous démontrons l’universalité des fluctuations dans l’intérieur du spectre dès lors que le second moment est fini.

MSC:60B20; 82B44

Keywords:Wigner matrix; Gaussian divisible; Optimal moment condition; Universality; Tracy–Widom distribution

1. Introduction and results

1.1. Introduction

An Hermitian Wigner matrix is a random Hermitian matrix with independent elements respecting the Hermitian symmetry. The local eigenvalue statistics of these random matrices is expected to be universal in the sense that it is independent of the distribution of the individual matrix elements, at least under suitable assumptions on the moments of the elements. There are two basic cases. We can either look in the bulk of the spectrum or at the edge around the largest eigenvalue. It is conjectured that, if we assume that the real and imaginary parts of the elements all have mean value zero, variance σ²>0 and that there is a uniform bound on the fourth moment, then the appropriately scaled eigenvalue point process at the edge should converge to the Airy kernel point process. Furthermore the largest eigenvalue should asymptotically fluctuate according to the Tracy–Widom distribution. This problem is still open, but there are results under stronger moment assumptions. The breakthrough result by Soshnikov, [20], showed that the result is true if the distribution is symmetric and has sub-gaussian tails. Soshnikov’s result is based on moment methods. The condition on the moments has been weakened to 18+εmoments (or 36+εmoments, see [1]) in [16].

In the bulk it is expected that the local eigenvalue point process converges to the sine-kernel point process. The exact conditions needed for this to be true are not clear. The result in the bulk was proved for a sub-class of Wigner

1Suported by the Swedish Research Council (VR) and the Knut and Alice Wallenberg Foundation KAW2010.0063.

matrices, so called Gaussian divisible Hermitian Wigner matrices in [14]. A Gaussian divisible Hermitian Wigner matrix is an Hermitian Wigner matrixWof the formW=X+√

κV, whereXis an Hermitian Wigner matrix andV an independent GUE matrix. In [14] it was assumed that the elements ofXhave uniformly bounded 6+εmoments.

Spectacular progress has recently been made on this problem by Tao and Vu, [23], with their four-moment theorem, and by Erdös, Ramirez, Schlein and H.-T. Yau using a different approach, [11]. Tao and Vu assume subexponential tails for the distribution of the matrix elements. Erdös, Ramirez, Schlein and H.-T. Yau make rather strong regularity assumptions on the distribution and parts of the argument use methods related to the approach in [14] and this paper.

A combined effort, [12], removed some of the assumptions in [23]. Thus, the universality result in the bulk is now established under the assumption of subexponential decay of the tails of the distributions of the matrix elements.²

Very recently, Tao and Vu, [22], also generalized Soshnikov’s result using an approach analogous to that in their paper on bulk universality. They obtain universality at the edge under the assumption of subexponential deacy and vanishing third moments. The result in this paper can be used to remove this third moment assumption, see Theo- rem1.5.

The four-moment theorem indicates that the class of Gaussian divisible Wigner matrices is a good testing ground for what we can expect for general Wigner matrices. In this paper we therefore return to the case of Gaussian divisible Hermitian Wigner matrices with the aim of establishing universality results within this class under weak moment conditions. In particular, we prove universality at the edge under the optimal assumption that the fourth moment is finite. It is known that if we have fewer than four moments then the behaviour around the largest eigenvalue is instead described by a Poisson process, see [1,8,21].

We also show universality in the bulk within the class of Gaussian divisible Hermitian Wigner matrices under the assumption that the second moment is finite. It is not clear that this is the optimal condition. Rather, close to the origin we may still expect sine-kernel universality even if the second moment is infinite, see [10].

The results are obtained using a development of the techniques in [14] which were based on a contour integral formula for a correlation kernel from [9]. In [14] an important tool was a concentration of measure estimate for the empirical eigenvalue distribution for a Wigner matrix from [13]. Here, due to the weak moment assumptions we have to proceed differently and in particular the choice of contours in the contour integral formula becomes more delicate.

Hence, the analysis that was done in [14] has to be modified in the technical details and this is somewhat subtle as can be expected since we are at the borderline of the validity of the conclusions of the theorem.

1.2. Results

We turn now to precise statements of our results. The n×n random matrix X is anHermitian Wigner matrixif X=(x_ij)is Hermitian, Rex_ij, Imx_ij, 1≤i < j≤n, andX_jj, 1≤j≤n, are all independent and satisfy

(i) E[Rexij] =E[Imxij] =0,1≤i≤j≤n,

(ii) E[(Rex_ij)²] =E[(Imxij)²] =σ²/2,1≤i < j≤n, (iii) E[x_jj²] =σ²,

where the varianceσ²<∞. The distribution of the different real and imaginary parts need not be identical and could depend onn. We will also assume that

(iv) limn→∞ 1 n²

1≤i≤j≤nE[|x_{j k}|²1(|x_{j k}|> η√ n)] =0

for any constantη >0. Here 1(A)denotes the inicator function for the eventA. This last condition is automatic if we have i.i.d. elements. Under assumptions (i)–(iv) we know that the semi-circle law holds, [5].

We will say thatW is aGaussian divisible Hermitian Wigner matrixif it can be written W=X+√

κV , (1.1)

whereXis an Hermitian Wigner matrix,κa positive constant andV an independent GUE-matrix. We take the GUE- measure to be

1

Z_ne^−trV2/2dV .

2Very recently [24] the assumption on the distribution has been reduced to a finite but large number of moments.

Without loss of generality we can choose the varianceσ²=1/4.

Let{λ_j}be the eigenvalues of√

nW. The sequence{λ_j/n}is asymptotically distributed according tothe Wigner semi-circle law, [5],

ρ(x)= 2 π(1+4κ)

(1+4κ−x²)₊. (1.2)

LetC_c(R)denote the set of all continuous functions with compact support, andC_c⁺(R)the subset ofC_c(R)of non- negative functions. Forb >0 let

K_sine^b (u, v)=sinb(u−v)

π(u−v) (1.3)

be thesine kernelwith densityb/π. Thesine-kernel point processon infinite point configurations{μj}on the real line is the determinantal point process defined by

E^bsine

exp

−

j

ψ (μ_j)

=det I−φ^1/2K_sine^b φ^1/2

(1.4) for allψ∈C⁺_c(R), whereφ=1−e^−ψ. Here, the right-hand side is the Fredholm determinant onL²(R)with kernel φ^1/2K_sine^b φ^1/2.

Theorem 1.1. LetW be a Gaussian divisible Hermitian Wigner matrix as in(1.1),whereXsatisfies the conditions (i)–(iv)and let{λj}be the eigenvalues of√

nW.Assume thatdn/n→d asn→ ∞,where|d|<√

1+4κ,and let β= 2

1+4κ

1+4κ−d². (1.5)

Then,

nlim→∞E

exp

− n

j=1

ψ (λ_j−d_n)

=E^β_sine

exp

−

j

ψ (μ_j)

(1.6) for allψ∈C_c⁺(R).

The theorem will be proved in Section 2.2. The theorem shows that the appropriately scaled eigenvalue point process converges weakly in the bulk, i.e. in the interior of the support of the semi-circle law, (1.2), to the sine kernel point process with density given by the semi-circle law. This theorem is an extension of the main result theorem in [14], see also [7].

We turn now to the edge behaviour. It is known that if the matrix elements are heavy-tailed with no fourth moment, then the eigenvalue point process at the edge converges to a Poisson point process with a certain density, see [1,8] and [21]. Thus, in order to get the same edge behaviour as for GUE we have to assume at least that the fourth moment is finite. It is known, see [3], that finite fourth moments is necessary and sufficient for the largest eigenvalue to converge to the edge of the support of the semi-circle. We will show that within the class of Gaussian divisible Wigner matrices finite fourth moments suffices for Tracy–Widom asymptotics.

The eigenvalue statistics of a GUE-matrix at the edge is described by the Airy kernel point process. TheAiry kernel is defined by

A(x, y)= _∞

0

Ai(x+t )Ai(y+t )dt=Ai(x)Ai(y)−Ai(x)Ai(y)

x−y . (1.7)

TheAiry kernel point processon infinite point configurations{μj}on the real line is the determinantal point process defined by

EAiry

exp

−

j

ψ (μj)

=det I−φ^1/2Aφ^1/2

(1.8)

for allψ∈C_c⁺(R), whereφ=1−e^−ψ. The Airy kernel point process has almost surely a last particleμ_maxwhose distribution is given by theTracy–Widom distribution,

PAiry[μ_max≤t] =F_TW(t)=det(I−A)_L2(t,∞). (1.9)

Here det(I−A)_L2(t,∞) is the Fredholm determinant of the trace-class operator onL²(t,∞) with integral kernel A(x, y).

We can now state our result on the edge statistics.

Theorem 1.2. LetWbe a Gaussian divisible Hermitian Wigner matrix, (1.1),wth finite fourth moments,i.e.there is a constantK <∞independent ofnsuch that

max

1≤i≤j≤nE

|x_ij|⁴

≤K. (1.10)

Let{λ_j}be the eigenvalues of√

nW,and let γ=√

1+4κ, δ=1 2

√1+4κ.

Then,

nlim→∞E

exp

− n

j=1

ψ (λj−γ n)/δn^1/3

=EAiry

exp

−

j

ψ (μj)

(1.11)

for allψ∈C_c⁺(R).Furthermore,ifλmax=max1≤j≤nλj,then

nlim→∞P

(λmax−γ n)/δn^1/3≤t

=FTW(t) (1.12)

for allt∈R.

The theorem will be proved in Section3.2.

Remark 1.3. When we have two but not four moments we have asymptotically the semi-circle law,the local eigenvalue statistics in the bulk is given by the sine-kernel point process,but the local eigenvalue statistics around the largest eigenvalue,which lies outside the semi-circle,is given by a Poisson process.It would be interesting to investigate the change in statistics as we move towards the edge.In terms of eigenvectors we should move from localized eigenvectors to de-localized eigenvectors.This problem is perhaps even more interesting when we have heavy-tailed distributions with unbounded variance.The global eigenvalue distribution is then no longer given by the semi-circle law and the scaling is different, [6].See[10]for a discussion.It is possible that the methods of the present paper could be extended to yield e.g.the sine-kernel point process close to the origin in this case also.This would probably require an improvement of the estimate(2.40),which still holds,but is not good enough.

Remark 1.4. When revising the present paper for publication we learnt about the papers[17]and[18].The results of[17]can be used to give another proof of Theorem1.1.That paper also uses the contour integral formula but the technical details are different.The very recent paper[18]gives alternative approach to Theorem1.2,again with the same starting point but different technical details.

As mentioned in the introduction Tao and Vu have recently extended the four-moment theorem to the edge, but since they compared with GUE they had to assume vanishing third moment. By combining with Theorem1.2we can see that the third moment condition is not necessary. We formulate this only for the fluctuations of the largest eigenvalue.

Theorem 1.5. Assume thatM=(m_ij)is an Hermitian Wigner matrix with subexponential decay,i.e.there are con- stantsC, C>0such that

P

|m_ij| ≥t^C

≤e^−t

for allt≥Cand all1≤i≤j≤n.Letλ_maxbe the largest eigenvalue of√

nM,and assume that the varianceσ²=1.

Then

nlim→∞P

(λ_max−2n)/n^1/3≤t

=F_TW(t) (1.13)

for allt∈R.

Proof. We can choose a Gaussian divisible Wigner matrixM so that the moments ofMandMmatch up to order three, see [23]. The result then follows from (1.12) and [22], Theorem 1.13; compare the proof of Theorem 1.16 in

[22].

2. Bulk universality

2.1. Convergence to the sine kernel point process

Consider nBrownian motions x1(t), . . . , xn(t) onRstarting at ν1, . . . , νn and conditioned never to intersect. The random positions at timeSthen form a determinantal point process with correlation kernel

K_n,S^ν (u, v)= 1 (2πi)²S

γ_L

dz

Γ_M

dwe^{(w2−2vw−z2+2uz)/2S} 1 w−z

n

j=1

w−νj

z−ν_j , (2.1)

whereν= {ν_j}ⁿ_j=1,γ_Lis the contour given by the positively oriented rectangle with corners at±L±i andΓ_M is the contour given bys→M+is, withM > L, see [14]. HereLis chosen so large that all the pointsν_jlie insideγ_L. Let E^ν denote the expectation with respect to the family of non-intersecting Brownian motions, and letφ∈C_c(R)satisfy 0≤φ≤1. Then,

E^ν _n

j=1

1−φ x_j(S)

=det I−φ^1/2K_n,S^ν φ^1/2

, (2.2)

where the right-hand side is a Fredholm determinant onL²(R)with respect to the finite rank kernelφ^1/2K_n,S^ν φ^1/2. This is useful for studying Gaussian divisible Wigner matrices because of the following fact. Let EX denote the expectation with respect to the Wigner matrixXand lety(X)= {y_j(X)}ⁿ_j=1be the eigenvalues of√

nX. Furthermore let EW denote the expectation with respect to the Gaussian divisible Wigner matrixW, (1.1). Then, [14], for ψ∈ C_c⁺(R),

EW

exp

− n

j=1

ψ (λ_j)

=EX

E^y(X)

exp

− n

j=1

ψ x_j(S_n)

, (2.3)

where{λ_j}are the eigenvalues ofWandS_n=κn. To use this formula we need good control of the kernelK_n,S^ν given by (2.1) for allν=y(X)except those in a set of negligible probability.

We can make a change of variablesz→Sz,w→Swin (2.1) to get K_n,S^ν (u, v)= 1

(2πi)²

γ_L

dz

Γ_M

dwe^{S(w2−z2)/2+uz−vw} 1 w−z

n

j=1

Sw−ν_j

Sz−ν_j (2.4)

withγ_LandΓ_M as above and where all theν_j/Slie insideγ_L. LetDbe a constant that could depend onν andS. It follows from (2.4) that

K_n,S^ν (u−SD, v−SD)= 1 (2πi)²

γL

dz

ΓM

dwe^{S(f (z)−f (w))+uz−vw} 1

w−z, (2.5)

where

f (z)=z²

2 +Dz+1 S

n

j=1

log(Sz−νj). (2.6)

We want to do a saddle point analysis asS→ ∞. The conditionf(a+ib)=0 gives the equations n

j=1

S

(Sa−ν_j)²+S²b² =1 (2.7)

and n

j=1

Sa−ν_j

(Sa−νj)²+S²b²+a+D=0. (2.8)

Note that if we take D=D(ν)=

n

j=1

νj

ν_j²+b²S², (2.9)

then we can takea=0 and letbbe the solution of n

j=1

S

ν_j²+b²S²=1. (2.10)

We can now show an approximation result forK_n,S^ν (u−SD(ν), v−SD(ν))in terms of the sine-kernel and this will suffice for our investigation in the bulk of Gaussian divisible Wigner matrices. Define, for a given setνand a positive numberS

Bn,S=

ν; there is ab >0 such that n

j=1

S

ν_j²+b²S² =1

. (2.11)

Hence, ifν∈B_n,S, there is a uniqueb=b(ν)such that (2.10) holds. Furthermore, define for forv∈B_n,S, A(ν)=

n

j=1

S³b²

(ν_j²+b²S²)². (2.12)

We have the following proposition.

Proposition 2.1. Ifν∈B_n,Sthere is a numerical constantCsuch that K_n,S^ν u−SD(ν), v−SD(ν)

−sinb(ν)(u−v) π(u−v)

≤ C

√SA(ν)e^3u2/SA(ν). (2.13)

Proof. We letf (z)be defined by (2.6) withD=D(ν). Let the contoursγ_±be given byγ_±:t→ ∓t±ib,t∈R, and Γ =Γ₀:s→is,s∈R. Set, withγ=γ₊+γ₋,

K˜_n,S^ν (u, v)= 1 (2πi)²

γ

dz

Γ

dwe^uz−vw 1

w−ze^{S(f (w)−f (z))}. (2.14)

We can deform the contourγ_Lto a rectangular contourγ_L with corners in±L±bi, and then move the contourΓ_M toΓ₀in the integral (2.4). We then pick up a contribution from the pole atw=zfor eachzonγ_L with Rez >0. The part ofγ_L with Rez >0 is a contour from−bi tobi and we can deform it to the straight line segment from−bi tobi.

Thus

K_n,S^ν u−SD(ν), v−sD(ν)

= 1 (2πi)²

γ_L

dz

Γ0

dwe^{S(w2−z2)+uz−vw} 1 w−z

n

j=1

Sw−ν_j Sz−νj

+ 1 2πi

bi

−bi

e^(u−v)zdz. (2.15)

We can now letL→ ∞and get, using (2.14), K_n,S^ν u−SD(ν), v−SD(ν)

−sinb(u−v)

π(u−v) = ˜K_n,S^ν (u, v). (2.16)

Hence, Theorem2.1, follows from K˜_n,S^ν (u, v)≤ C

√SA(ν)e^3u2/SA(ν) (2.17)

for all v∈B_n,S. In order to prove this inequality we have to choose the right contours in (2.14). The following computation motivates the choice of contours.

Letz(t )=x(t)+iy(t)and setg(t)=Ref (z(t )). Then, using (2.9) and (2.10) we see that g=

n

j=1

S(xx−yy)+xν_j

ν²_j+b²S² +S(xx+yy)−xν_j (Sx−ν_j)²+S²y²

. (2.18)

If we write the sum of the two fractions in (2.18) as one fraction the numerator becomes S²

−x²x+2xyy+y²x−b²x

ν_j+S³ xx−yy x²+y²

+b² xx+yy . We try to choosez(t )so that the expression in the numerator is independent ofνj. This gives

d dt

−1

3x³+y²x−b²x

=0 or

x

−1

3x²+y²−b²

=C.

Ifx(0)=0,y(0)= ±bwe getC=0 and two possibilitiesz(t )=i(t±b)orz(t )=t±i

t²/3+b². If we takez(t )=i(t±b)we get

d

dt Ref z(t )

= −St n

j=1

S²(t±b)(t±2b)

(ν_j²+b²S²)(ν_j²+(t+b)²S²). (2.19)

If instead we takez(t )=t±i

t²/3+b²we obtain d

dtRef z(t )

=St n

j=1

8S²t²/9+2b²S²

(ν_j²+b²S²)((St−ν_j)²+(t²/3+b²)S²). (2.20) Using this result we can prove

Lemma 2.2. Letw_±(s)=i(s±b)andz_±(t )=t±i

t²/3+b².Assume thatν∈B_n,S. (i) If±s+b≥0,then

Ref w_±(s)

−f (±bi)

≤ −1

6A(ν)s². (2.21)

(ii) For eacht∈R, Ref (±bi)−f z_±(t )

≤ −1

6A(ν)t². (2.22)

Proof. We see that, for−b≤s≤0, Ref w₊(s)

−f (bi)

=S³ ₀

s

t n

j=1

(t+b)(t+2b)

(ν_j²+b²S²)(ν_j²+(b+t )²S²)dt

≤S³ 0

s

t n

j=1

(b+t )b (ν_j²+b²S²)²dt

=A(ν) b

−s² 3

3 2b+s

≤ −A(ν) 6 s². Ifs≥0, we get

Ref w₊(s)

−f (bi)

=S³ _s

0

t n

j=1

(t+b)(t+2b)

(ν²_j+b²S²)(ν_j²+(b+t )²S²)dt

≤ − _s

0

t n

j=1

S³(t+b)²

(ν_j²+b²S²)(ν_j²+(b+t )²S²)dt.

If we use the fact thatx→x²(ν²+x²)⁻¹is increasing inx≥b, we see that the last expression is

≤ −A(ν) s

0

tdt= −1 2A(ν)s².

The contourw₋(s)is treated analogously. This proves (i) in the lemma.

Now, fort≥0, Ref z₊(s)

−f (bi)

=S t

0

τ n

j=1

8S²τ²/9+2b²S²

(ν_j²+b²S²)((Sτ−νj)²+S²(τ²/3+b²))dτ

≥S _t

0

τ n

j=1

8S²τ²/9+2b²S²

(ν_j²+b²S²)(2ν_j²+7S²τ²/3+b²S²).

It is easy to see that 8S²τ²/9+2b²S² 2ν²_j+7S²τ²/3+b²S²≥1

3 S²b² ν²_j+b²S²

and hence we obtain (2.22) forz₊(t)andt≥0. The argument fort≤0 and the argument forz₋(t)are similar.

We can now prove the estimate (2.17). Let γ₊ be given by z₊(−t ),t∈R,γ₋ by z₋(t),t∈R,Γ₊ byw₊(s), s≥ −b, andΓ₋byw₋(s),s≤b, wherez_±andw_±are as in Lemma2.2. Then,

K˜_n,S^ν (u, v)= 1 (2πi)²

γ₊+γ₋

dz

Γ₊+Γ₋

dwe^uz−vw 1

w−ze^{S(f (w)−f (z))}.

Consider the case whenzlies onγ₊andwonΓ₊. The other cases are similar. By Lemma2.2 1

(2πi)²

γ₊

dz

Γ₊

dwe^uz−vw 1

w−ze^{S(f (w)−f (z))}

≤ 1 4π²

_∞

−∞dt _∞

−b

ds e^ut

t²+(b+s−

t²/3+b²)²

e^{−SA(ν)(s2+t2)/6}. (2.23)

Sincet²+(b+s−

t²/3+b²)²≥(t²+s²)/3, we see that the expression in the right-hand side of (2.23) is

≤

√2 4π²

R²

e^ut

√t²+s²e^{−SA(ν)(s2+t2)/6}≤ C

SA(ν)e^3u2/A(ν),

whereCis a numerical constant. This completes the proof of Proposition2.1.

Assume now that we have a probability measurePνwith expectationEνon the point configurationsν= {ν_j}. We can then define a point processμ= {μ_j}ⁿ_j=1onRdepending onSby

En,S

_n

j=1

1−φ(μ_j)

=Eν

E^ν

_n

j=1

1−φ x_j(S)

(2.24) for everyφ∈C_c(R)with 0≤φ≤1.

We can now state the following proposition on convergence to the sine kernel point process defined by (1.4).

Proposition 2.3. Letα_n,β_n,δ_n,ω_nandS_nbe sequences such thatS_n>0,ω_n→ ∞,ω_n/log(Snα_n)→0andβ_n→ β >0asn→ ∞.Define,withS=S_nin the above definitions,

C_n=

ν∈B_n,Sn;A(ν)≥α_n,b(ν)−β_n≤1/ωn,D(ν)−δ_n≤

ω_nα_n/S_n

. (2.25)

Assume that the sequences can be chosen in such a way that

nlim→∞Pν[Cn] =1. (2.26)

Then,

nlim→∞En,S_n

exp

− n

j=1

ψ (μj+Snδn)

=E^β_sine

exp

−

j

ψ (μj)

(2.27)

for everyψ∈C_c⁺(R).

Proof. It is clear from (2.24) and (2.26) that it is enough to prove that

nlim→∞Eν

1CnE^ν

exp

− n

j=1

ψ x_j(S_n)+S_nδ_n

=E^β_sine

exp

−

j

ψ (μ_j)

. (2.28)

Here 1Adenotes the indicator function for the eventA. Writeφ=1−e^−ψ. Consider a fixedν∈C_nand write L^ν_n(u, v)=K_n,S^ν

n u−S_nD(ν), v−S_nD(ν) and

φ_n(u)=φ u+S_nδ_n−S_nD(ν) . It follows from (2.13) that

φ_n^1/2(u)L^ν_n(u, v)φ_n^1/2(v)−φ_n^1/2(u)K_sine^b(ν)φ^1/2_n (v)

≤ C

√S_nA(ν)e^Cu2/SnA(ν)φ_n^1/2(u)φ_n^1/2(v). (2.29)

There is a constantC such thatφ_n(u)=0 if|u| ≥S_n|D(ν)−δ_n| +C. Hence,φ_n(u)=0 if |u| ≥2√

ω_nα_nS_n forn large sinceν∈C_n. If|u| ≤2√

ω_nα_nS_n, then

√ C

S_nA(ν)e^Cu2/SnA(ν)≤ C

√S_nA(ν)e^Cωn≤ C (S_nα_n)^1/4 fornlarge, sinceω_n/log(Snα_n)→0 asn→ ∞. Thus, by (2.29)

φ_n^1/2(u)L^ν_n(u, v)φ_n^1/2(v)−φ_n^1/2(u)K_sine^b(ν)φ^1/2_n (v)≤ C

(S_nα_n)^1/4φ_n^1/2(u)φ^1/2_n (v) for allu, v. For a givenε >0 we thus have

φ_n^1/2(u)L^ν_n(u, v)φ_n^1/2(v)−φ_n^1/2(u)K_sine^β φ_n^1/2(v)≤εφ^1/2_n (u)φ_n^1/2(v) (2.30) for all sufficiently largenuniformly inν∈C_n, since|b(ν)−β_n| ≤1/ωnandβ_n→β asn→ ∞.

IfAis an operator onL²(R)with integral kernelA(x, y)then the Hilbert–Schmidt norm ofAis give byA²₂=

R²|A(x, y)|²dxdy. We now use the following lemma.

Lemma 2.4. IfAandB are trace class operators onL²(R)then det(I−A)−det(I−B)

≤ A−B2e^{−trA+(A−B2+2B2+1)2/2}+e^{(B2+1)2/2−trB} e^{−(trA−trB)}−1

. (2.31)

The lemma is proved in Section3.4.

It follows from (2.2) and a translation of variables that E^ν

exp

− n

j=1

ψ x_j(S_n)+S_nδ_n

=det I−φ_n^1/2L^ν_nφ_n^1/2

. (2.32)

Using (2.30), (2.31) and the fact that the sine kernel is translation invariant it is now straightforward to see that det I−φ_n^1/2L^ν_nφ_n^1/2

−det I−φ^1/2K_sine^β φ^1/2→0

uniformly forν∈C_nasn→ ∞. This completes the proof by (2.28) and (2.32).

2.2. Proof of bulk universality

In this section we will prove Theorem1.1on bulk universality for Gaussian divisible Hermitian Wigner matrices with finite second moment using Proposition2.3. Define

mn(z)=1 n

n

j=1

1 y_j−z=1

ntr X/√

n−z₋1

(2.33) for Imz=0. Then

EX

m_n(z)

→m(z)= −2z+

z²−1 (2.34)

asn→ ∞, [5], for eachz∈Cwith Imz=0. Letδ+βi,β >0, be given by m d+κ(δ+βi)

=δ+βi, (2.35)

which gives δ= − 2d

1+4κ, β= 2 1+4κ

1+4κ−d². (2.36)

Lemma 2.5. There is a sequenceδ_n+β_ni,β_n>0,such that EX

m_n d_n/n+κ(δ_n+β_ni)

=δ_n+β_ni (2.37)

andδ_n+β_ni→δ+βiasn→ ∞.

Proof. Defineg_n(z)=EX[m_n(d_n/n+κz)] −z. Theng_nis analytic in Imz >0. Since m_n(d_n/n+κz)−m_n(d+κz)≤|d_n/n−d|

(κImz)²

anddn/n→d asn→ ∞, it follows from (2.34) thatgn(z)→g(z)=m(d+κz)−zuniformly on compact subsets of Imz >0 asn→ ∞(by Montel’s theorem). Sinceg(δ+βi)=0 by (2.35) it follows by Hurwitz’ theorem that there is a sequenceδn+β_ni such thatg_n(δ_n+β_ni)=0 andδ_n+β_ni→δ+βi.

Set

c_n=d_n/S_n+δ_n, ν_j=y_j−c_nS_n. (2.38)

The probability measure onXinduces a probabilty measure onν= {νj}that we denote byPν. Now, using (2.1), we see that

K_n,S^y

n(u+c_nS_n, v+c_nS_n)=e^{((u+cnSn)2−(v+cnSn)2+v2−u2)/2Sn}K_n,S^ν

n(u, v) and from this it follows that

E^y

exp

− n

j=1

ψ x_j(S_n)−d_n

=E^ν

exp

− n

j=1

ψ x_j(S_n)+δ_nS_n .

Hence, EW

exp

− n

j=1

ψ (λj−dn)

=Eν

E^ν

exp

− n

j=1

ψ xj(Sn)+Snδn

. (2.39)

Choose α_n=α >0 fixed, to be specified below, β_n and δ_n as in Lemma 2.5, S_n=κn and ω_n=√

logn. Then Theorem1.1follows if we can show thatPν[C_n] →1 asn→ ∞withC_nas in (2.25).

To prove this we will use

Lemma 2.6. For eachz∈CwithImz=0we have the estimate EXm_n(z)−EX

m_n(z)²

≤ 2

n|Imz|². (2.40)

This is proved in [2]. For convenience we give the proof in Section3.4.

Define

M_n(τ )=m_n(κc_n+κτi)−EX

m_n(κc_n+κτi) . Note that, by (2.33) and (2.38)

m_n(κc_n+z)=1 n

n

j=1

1

ν_j/n−z. (2.41)

Set V_n=

ν;M_n(τ )≤ ω_n

n forτ =β_n, β/2,2βand 3β

. The result we need now follows from

Lemma 2.7. The following statements hold.

(i) Pν[Vn] →1asn→ ∞.

(ii) There is an α >0 such that if we choose αn=αand the other sequences as above,thenVn⊆Bn,S_n and Vn∩Bn,S_n⊆Cn,ifnis large enough.

Proof. Letτ >0 be fixed. Then by Chebyshev’s inequality and Lemma2.6 Pν

Mn(τ )>

ω_n n

≤ n

ω_nEXm_n(κcn+κτi)−EX

mn(κcn+κτi)²

≤ 2

ω_nκ²τ² →0,

asn→ ∞. We can apply this toτ =βn, β/2,2βand 3β noting thatβn≥β/2 ifnis large enough. This proves (i).

Note that

Remn(κc_n+κτi)= n

j=1

ν_j

ν_j²+τ²S_n² (2.42)

and

Immn(κcn+κτi)= n

j=1

S_nτ

ν_j²+τ²S_n². (2.43)

Furthermore, h(τ )=1

τ Imm(x+iτ )= 2 π

₁

−1

√1−t² (t−x)²+τ²dt is strictly decreasing inτ for each fixedx.

Define U_n=

ν;

n

j=1

Sn

ν_j²+4β²S²_n <1<

n

j=1

Sn

ν_j²+β²S_n²/4

.

We want to show thatVn⊆Unifnis large enough. Sinceh(τ )is strictly decreasing, (2.35) gives 1

2β Imm(κc+2κβi) <1−ε <1= 1

βImm(κc+κβi) <1+ε < 2

βImm(κc+κβi/2), if we chooseεsmall enough. Herec=d/κ+δ=limn→∞cn. It follows from this and (2.34) that

1 2β ImEX

m_n(κc_n+2κβi)

≤1−ε <1+ε≤ 2 βImEX

m_n(κc_n+κβi/2)

for allnlarge enough. Ifν∈V_nit follows from this that 1

2β Imm_n(κc_n+2κβi)≤1−ε+ ωn

n <1<1+ε− ωn

n ≤ 2

βImm_n(κc_n+κβi/2), and we see from (2.43) that this givesν∈Un.

Hence, ifnis large enough, then

β/2≤b(ν)≤2β (2.44)

for allν∈V_n. Letν∈V_n. Then, using (2.44), we see that A(ν)=

n

j=1

S_n³b²

(ν_j²+b²S_n²)²≥1 4

n

j=1

S_n³β² (ν_j²+4β²S_n²)²

≥ S_n 20

n

j=1

5S²_nβ²

(ν_j²+4β²S²_n)(ν_j²+9β²S_n²)= 1 20

_n

j=1

S_n ν_j²+4β²S_n²−

n

j=1

S_n ν²_j+9β²S_n²

.

By (2.34), (2.43) and the fact thatν∈V_nit follows from this that A(ν)≥ 1

20 1

2βImM_n(2β)− 1

3βImM_n(3β)

+ 1 20

1 2βImEX

m_n(κc_n+2κβi)

− 1 3β ImEX

m_n(κc_n+3κβi)

≥ 1 40

1

2βImm(κc+2κβi)− 1

3βImm(κc+3κβi) .

=α >0 for largen.

Next, we will show that, ifnis large enough, b(ν)−β_n≤C

ω_n n ≤ 1

ω_n (2.45)

for allν∈V_n. It follows from (2.10), (2.37), (2.43) andν∈V_n, that

n

j=1

S_n ν_j²+β_n²S_n²−

n

j=1

S_n ν_j²+b²S²_n

≤

ω_n n ,